Riesz representation theorems for vector lattices and Banach lattices of regular operators
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| Format: | Preprint |
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2025
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| _version_ | 1866914462157504512 |
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| author | de Jeu, Marcel Jiang, Xingni |
| author_facet | de Jeu, Marcel Jiang, Xingni |
| contents | For a non-empty locally compact Hausdorff space $X$ and a Dedekind complete normal vector lattice $E$, we show that the vector lattice of norm to order bounded operators from ${\text C}_{\text c}(X)$ or ${\text C}_0(X)$ into $E$ is isomorphic to the vector lattice of $E$-valued regular Borel measures on $X$. When $E$ is an order continuous Banach lattice, the isomorphism is an isometric isomorphism between Banach lattices. When $X$ is compact, every regular operator from $\mathrm{C}(X)$ into $E$ is norm to order bounded. For some spaces $E$, such as KB-spaces or the regular operators on a KB-space, every regular operator from ${\mathrm C}_0(X)$ into $E$ is norm to order bounded. Additional results are obtained for the whole space of regular operators from ${\text C}_{\text c}(X)$ into an order continuous Banach lattice.
As a preparation, vector lattices and Banach lattices, resp. cones, of measures with values in a Dedekind complete vector lattice $E$, resp. in the extended positive cone of $E$, are investigated, as well as vector and Banach lattices of norm to order bounded operators.
When $E$ is the real numbers, our results specialise to the well-known Riesz representation theorems for the order and norm duals of ${\text C}_{\text c}(X)$ and ${\text C}_0(X)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_12568 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Riesz representation theorems for vector lattices and Banach lattices of regular operators de Jeu, Marcel Jiang, Xingni Functional Analysis Primary 47B65, Secondary 28B15, 46A40, 46B42 For a non-empty locally compact Hausdorff space $X$ and a Dedekind complete normal vector lattice $E$, we show that the vector lattice of norm to order bounded operators from ${\text C}_{\text c}(X)$ or ${\text C}_0(X)$ into $E$ is isomorphic to the vector lattice of $E$-valued regular Borel measures on $X$. When $E$ is an order continuous Banach lattice, the isomorphism is an isometric isomorphism between Banach lattices. When $X$ is compact, every regular operator from $\mathrm{C}(X)$ into $E$ is norm to order bounded. For some spaces $E$, such as KB-spaces or the regular operators on a KB-space, every regular operator from ${\mathrm C}_0(X)$ into $E$ is norm to order bounded. Additional results are obtained for the whole space of regular operators from ${\text C}_{\text c}(X)$ into an order continuous Banach lattice. As a preparation, vector lattices and Banach lattices, resp. cones, of measures with values in a Dedekind complete vector lattice $E$, resp. in the extended positive cone of $E$, are investigated, as well as vector and Banach lattices of norm to order bounded operators. When $E$ is the real numbers, our results specialise to the well-known Riesz representation theorems for the order and norm duals of ${\text C}_{\text c}(X)$ and ${\text C}_0(X)$. |
| title | Riesz representation theorems for vector lattices and Banach lattices of regular operators |
| topic | Functional Analysis Primary 47B65, Secondary 28B15, 46A40, 46B42 |
| url | https://arxiv.org/abs/2508.12568 |